Theory of General Invariance

Postulate I: $[\hat{x}_{i}, \hat{x}_{j}] = il_{p}^{2}\hat{E}$ where $\hat{E}$ is a 0-1 adjacency matrix (which we call the entanglement field operator).

Postulate 3: The operator-valued power spectrum $ \mathcal{P}[\hat{A}(\mathcal{G})] \equiv |G> $ generates a locally finite-dimensional Hilbert space with the metric eigenfunction $ |g> = |g_{+}> + |g_{-}> $.

Postulate 2: For every linear self-adjoint operator $ \hat{A} $ there exists an underlying graph $ \mathcal{G} $.

Postulate 2a: When spacetime is in superposition, the vertices $ (e_{1}, e_{2} ) $ of $ \mathcal{G} $ are elements of the Geometric (Clifford) algebra $ Cl(\mathbb{R}, 2) $; the imaginary constant is a bivector (area element) $ i = e_{1}e_{2} $.

Postulate 2b: When spacetime is not in superposition, the vertices $ (e_{1}, e_{2}, e_{3} )$ of $ \mathcal{G} $ are elements of the Geometric (Clifford) algebra $ Cl(\mathbb{R}, 3) $; the imaginary constant is a trivector (volume element) $ i = e_{1}e_{2}e_{3} $.

The geometrized phase space is $ (\hat{p} \cdot \sigma, \hat{q} \cdot \sigma )$ (which we call qubitspace).

The geometrized phase space is $ ( \hat{p} \otimes \hat{p}, \hat{q} \otimes \hat{q}) $ (which we call bitspace).

The momentum $ \hat{p} = -e_{1}e_{2}\hbar \nabla $ is an operator-valued bivector (area element).

The position $ \hat{q} = \hat{x}e_{1} + \hat{y}e_{2} + \hat{z}e_{3} $ is an operator-valued vector (invariant w.r.t basis change).

The momentum $ \hat{p} = -e_{1}e_{2}e_{3}\hbar \nabla $ is an operator-valued trivector (volume element).

The position $ \hat{q} = \hat{x}e_{1} + \hat{y}e_{2} + \hat{z}e_{3} $ is an operator-valued vector (invariant w.r.t basis change).

In the $ Cl(\mathbb{R}, 3) $ basis $ e_{1}e_{2}e_{3}l_{p}^{2}\hat{E} $ is a matrix-valued trivector (volume element).

In the $ Cl(\mathbb{R}, 2) $ basis $ e_{1}e_{2}l_{p}^{2}\hat{E} $ is a matrix-valued bivector (area element).

All QM operators become adjacency operators $ \hat{A}(\mathcal{G}) $ with $ \mathcal{G} $ specifying the microscopic structure of spacetime.

SU(2) X SU(2)

SO(3) X SO(3)

Generalized Uncertainty Principle: $ \sigma_{i}\sigma_{j} \geq \dfrac{l_{p}^{2}}{2}|<\hat{E}>|^{2} $ where $ 0 \leq \ <\hat{E}> \ \leq 1 $.

The Entanglement Field Equation: $ |G> = |g_{+}> + |g_{-}> $

emergent arrow of time: $ <\hat{E}> \longrightarrow 0 $ (sparse limit).

Smooth manifold in the sparse limit.

Holographic Flow: Transitions between the $ Cl(\mathbb{R}, 3) $ and $ Cl(\mathbb{R}, 2) $ basis generates the $ SO(4) \longrightarrow SO(3) \times SO(3) $ $ SU(2) \times SU(2) \longrightarrow SO(4) $ representation of 4-dim Euclidean space $ \mathbb{R}^{4} $.

$ \mathcal{G} $ has two unique states: superimposed (degrees of freedom optimally compressed) and collapsed (optimally decompressed).

Emergent Time

Principle of Least Description: Nature will choose either a smooth manifold or a Hilbert space as an underlying mathematical structure.

Principle of Scale Invariance: the laws of physics do not depend on scale.

Principle of Finiteness: the laws of physics strictly forbid infinite quantities.

Principle of General Invariance: the laws of physics are scale-invariant within a fundamental UV/IR regime.

Principle of Least Description + Principle of Finiteness + Principle of Scale Invariance = Principle of General Invariance